Bulletin of the American Physical Society
69th Annual Meeting of the APS Division of Fluid Dynamics
Volume 61, Number 20
Sunday–Tuesday, November 20–22, 2016; Portland, Oregon
Session L8: Nonlinear Dynamics: Model Reduction |
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Chair: Maciej Balajewicz, University of Illinois at Urbana-Champaign Room: B116 |
Monday, November 21, 2016 4:30PM - 4:43PM |
L8.00001: Lagrangian dimensionality reduction of convection dominated nonlinear flows Maciej Balajewicz, Rambod Mojgani We introduce a new projection-based model reduction approach for convection dominated nonlinear fluid flows. In this method the evolution of the fluid is approximated in the Lagrangian frame of reference. More specifically, global basis functions are utilized for both the state of the system and the positions of the Lagrangian computational domain. In this approach, wave-like solutions exhibit low-rank structure and thus, can be approximated efficiently using a small number of reduced bases. The proposed approach is successfully demonstrated for the reduction of several simple but representative flow problems. [Preview Abstract] |
Monday, November 21, 2016 4:43PM - 4:56PM |
L8.00002: Novel Stochastic Mode Reduction For General Irreversible Systems Markus Schmuck, Marc Pradas, Grigorios A. Pavliotis, Serafim Kalliadasis We outline a novel stochastic mode reduction strategy for nonlinear irreversible dynamical systems. Our methodology is based on the concept of maximum information entropy together with spectral characteristics of linear operators and a dynamic renormalization strategy [1,2]. It results in low-dimensional stochastic equations equipped with a systematically determined noise term. We demonstrate the performance and validity of our novel method with various physical model prototypes such as front propagation in reaction diffusion systems, phase separation in binary mixtures, and coarsening of interfaces. These are just a few examples demonstrating the wide applicability of our computational mode reduction. \\ 1. M. Schmuck, M. Pradas, S. Kalliadasis \& G.A. Pavliotis, Phys. Rev. Lett. 110:244101 2013. \\ 2. M. Schmuck, M. Pradas, G.A. Pavliotis \& S. Kalliadasis, IMA J.Appl. Math. 80:273-301 2015. [Preview Abstract] |
Monday, November 21, 2016 4:56PM - 5:09PM |
L8.00003: Computing Finite-Time Lyapunov Exponents with Optimally Time Dependent Reduction Hessam Babaee, Mohammad Farazmand, Themis Sapsis, George Haller We present a method to compute Finite-Time Lyapunov Exponents (FTLE) of a dynamical system using Optimally Time-Dependent (OTD) reduction recently introduced by H. Babaee and T.P. Sapsis (\emph{A minimization principle for the description of modes associated with finite-time instabilities}, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, Vol. 472, 2016). The OTD modes are a set of finite-dimensional, time-dependent, orthonormal basis $\{ \mathbf{u}_i(\mathbf{x},t)\}|_{i=1}^N$ that capture the directions associated with transient instabilities. The evolution equation of the OTD modes is derived from a minimization principle that optimally approximates the most unstable directions over finite times. To compute the FTLE, we evolve a single OTD mode along with the nonlinear dynamics. We approximate the FTLE from the reduced system obtained from projecting the instantaneous linearized dynamics onto the OTD mode. This results in a significant reduction in the computational cost compared to conventional methods for computing FTLE. We demonstrate the efficiency of our method for double Gyre and ABC flows. [Preview Abstract] |
Monday, November 21, 2016 5:09PM - 5:22PM |
L8.00004: Evaluating the accuracy of the dynamic mode decomposition Hao Zhang, Scott Dawson, Clarence Rowley, Eric Deem, Louis Cattafesta Dynamic mode decomposition (DMD) is a practical way to extract dynamic information about a fluid flow directly from data. As a data-driven method, DMD can suffer from error, which can be difficult to quantify without knowledge of an exact solution, free from noise or external disturbances. Here we propose an evaluation metric for the accuracy of DMD results (eigenvalues, modes, and eigenfunctions), by exploiting a connection between DMD and the Koopman operator, a linear operator acting on functions of the flow state. In particular, a DMD mode is considered "accurate" if the corresponding eigenfunction closely approximates a Koopman eigenfunction. With this definition, we can assess the accuracy of any individual DMD mode directly from data, without requiring the direct calculation of the Koopman operator. We demonstrate the use of this criterion with a range of examples including synthetic, numerical, and experimental data. [Preview Abstract] |
Monday, November 21, 2016 5:22PM - 5:35PM |
L8.00005: Sparse Identification of Nonlinear Dynamics (SINDy) Steven Brunton, Joshua Proctor, Nathan Kutz This work develops a general new framework to discover the governing equations underlying a dynamical system simply from data measurements, leveraging advances in sparsity techniques and machine learning. The so-called sparse identification of nonlinear dynamics (SINDy) method results in models that are parsimonious, balancing model complexity with descriptive ability while avoiding over fitting. The only assumption about the structure of the model is that there are only a few important terms that govern the dynamics, so that the equations are sparse in the space of possible functions; this assumption holds for many physical systems in an appropriate basis. We demonstrate the algorithm on a wide range of problems, from simple canonical systems, including the chaotic Lorenz system, to the canonical fluid vortex shedding behind an circular cylinder at Re=100. We also show that this method generalizes to parameterized systems and systems that are time-varying or have external forcing. With abundant data and elusive laws, data-driven discovery of dynamics will continue to play an increasingly important role in the characterization and control of fluid dynamics. [Preview Abstract] |
Monday, November 21, 2016 5:35PM - 5:48PM |
L8.00006: A Low-Order Galerkin Model Based on DMD and Adjoint-DMD modes Wei Zhang, Mingjun Wei Dynamic Mode Decomposition (DMD) has emerged as a new tool for the understanding of flow dynamics associated with frequencies. The DMD modes computed by this process have been considered as an alternative of base functions for model order reduction. However, DMD modes are not orthogonal bases which are usually desired for the simplicity of Galerkin models. Therefore, we used the bi-orthogonal pair of DMD modes and adjoint DMD modes to solve this problem, and introduced an easy approach to derive a simple DMD-Galerkin projection model. The introduction of adjoint DMD modes also provides an easy way to rank DMD modes for order reduction. The approach is applied on a flow-passing-cylinder case in both transition and periodic stages. For the periodic case, DMD-Galerkin model is similar to POD-Galerkin model; and for the transition case, DMD-Galerkin model carries more clear frequency features. [Preview Abstract] |
Monday, November 21, 2016 5:48PM - 6:01PM |
L8.00007: Model Order Reduction for Fluid Dynamics with Moving Solid Boundary Haotian Gao, Mingjun Wei We extended the application of POD-Galerkin projection for model order reduction from usual fixed-domain problems to more general fluid-solid systems when moving boundary/interface is involved. The idea is similar to numerical simulation approaches using embedded forcing terms to represent boundary motion and domain change. However, such a modified approach will not get away with the unsteadiness of boundary terms which appear as time-dependent coefficients in the new Galerkin model. These coefficients need to be pre-computed for prescribed motion, or worse, to be computed at each time step for non-prescribed motion. The extra computational cost gets expensive in some cases and eventually undermines the value of using reduced-order models. One solution is to decompose the moving boundary/domain to orthogonal modes and derive another low-order model with fixed coefficients for boundary motion. Further study shows that the most expensive integrations resulted from the unsteady motion (in both original and domain-decomposition approaches) have almost negligible impact on the overall dynamics. Dropping these expensive terms reduces the computation cost by at least one order while no obvious effect on model accuracy is noticed. [Preview Abstract] |
Monday, November 21, 2016 6:01PM - 6:14PM |
L8.00008: Sensor Placement in Multiscale Phenomena using Multi-Resolution Dynamic Mode Decomposition Krithika Manohar, Eurika Kaiser, Steven L. Brunton, J. Nathan Kutz Multiscale processes pose challenges in determining modal decompositions with physical meaning which can render sensor placement particularly difficult. Localized features in space or time may play a crucial role for the phenomenon of interest but may be insufficiently resolved due to their low energy contribution. We consider optimal sensor placement using spatial interpolation points within the framework of data-driven modal decompositions. In recent years, the Discrete Empirical Interpolation Method or DEIM and variants like QDEIM have gained popularity for interpolating nonlinear terms arising in model reduction using Proper Orthogonal Decomposition modes. We extend this sensor placement approach to multiscale physics problems using Multi-Resolution Dynamic Mode Decomposition or mrDMD (Kutz et al., 2015), an unsupervised multi-resolution analysis in the time-frequency domain that separates flow features occurring at different timescales. The discovered sensors achieve accurate flow state reconstruction in representative multiscale examples including global ocean temperature data with an energetic El Ni\~{n}o mode. Interestingly, this method places sensors near coastlines without imposing additional constraints, which is beneficial from an engineering perspective. [Preview Abstract] |
Monday, November 21, 2016 6:14PM - 6:27PM |
L8.00009: Compressed sensing DMD with control Zhe Bai, Eurika Kaiser, Joshua Proctor, J. Nathan Kutz, Steven Brunton The dynamic mode decomposition (DMD) has been widely adopted in the fluid dynamics community, in part due to its ease of implementation, its connection to nonlinear dynamical systems, and its highly extensible formulation as a linear regression. This work combines the recent innovations of compressed sensing DMD and DMD with control, resulting in a new computational framework to extract spatiotemporal coherent structures using subsampled data from a complex system with inputs or control. The resulting compressed DMD with control (cDMDc), has two major uses in high-dimensional systems, such as a fluid flow: 1) if only subsampled or compressive measurements are available, it is possible to used compressed sensing to reconstruct full-dimensional DMD modes, and 2) if full data is available, it is possible to accelerate computations by first pre-compressing data and then reconstructing full modes from compressed DMD computations. In both cases, the addition of DMD with inputs and control makes it possible to disambiguate the natural unforced dynamics from the effect of actuation. We demonstrate this architecture on a number of relevant examples from fluid dynamics. [Preview Abstract] |
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